Spin-Orbit Coupling Effects in Bilayer and Optical Lattice Systems 纪安春 首都师范大学物理系

Collaborator Qing Sun (首都师范大学) Lin Wen (重庆师范大学) W.-M. Liu (物理所) Liang He & Walter Hofstetter (Goethe–Universität, Germany) G. Juzeliunas (Vilnius University, Lithuania)

Outline I. Bilayer Spin-Orbit Coupling II. Optical Lattices subjected to Spin-Orbit Coupling 1) SO coupled –U Hubbard model 2) SO coupled +U Hubbard model

I. Bilayer Spin-Orbit Coupling 1D SOC实验实现 Effective single particle Hamiltonian： Y.-J. Lin et al., Nature 471, 83 (2011) J.-Y. Zhang et al., PRL 109, 115301 (2012) P. J. Wang et al., PRL 109 085303 (2012) L. W. Cheuk et al., PRL 109,095302 (2012)

2D SOC：a Challenge! Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark States J. Ruseckas et al., PRL 95, 010404 (2005) \omega_1,2,3 表示的是一束raman光，角度是theta和phi,他分别在x,y,z三个方向进行投影。 Tripod-coupling scheme

Drawback： The two dark states are not the ground states!

Realistic Rashba and Dresselhaus spin-orbit coupling for neutral atoms D. L. Campbell, G. Juzeliunas, and I. B. Spielman, PRA 84, 025602 (2011) Drawback： i) The total phase should be fixed to be \pi! ii) The four internal states are not easy to be Implemented.

Experimental realization of a two-dimensional synthetic spin-orbit coupling in ultracold Fermi gases Jing Zhang group, arXiv:1506.02861 Three Energy level scheme: Transition loop:

Challenge： i) Gap of the Dirac cone can not be opened. ii) many-body physics is a challenge.

Bilayer Spin-Orbit Coupling Q. Sun, L. Wen, W.-M. Liu, G. Juzeliunas, and A.-C. Ji, PRA 91, 033619 (2015)

Energy level loop： In plane mometum: Effective single particle Hamiltonian：

Single-particle spectrum Lowest branch Many-body Ground States Variation Ansatz

Ground state phase diagram PW: plane-wave SP-I: normal stripe SP-II: time-breaking stripe FSL: fractionalized skyrmion lattice

SP-I: Conventional stripe SP-II: Time-symmetry breaking stripe

FSL: Fractional Skyrmion lattices

Drawback： i) 2D SOC, but not the Rashba-type SOC! ii) The experimental scheme is not easy!

Further Improvements： i) Gap of the Dirac cone can be easily opened. ii) The scheme includes only two internal states, which is easy for many-body physics. Relevant results are in preparation!

II. Optical Lattices subjected to Spin-Orbit Coupling K. Osterloh et al. PRL 95, 010403 (2005).

Non-Abelian gauge potentials acting on cold atoms in optical lattices K. Osterloh et. al., PRL 95, 010403 (2005)

1) SO coupled –U Fermi-Hubbard model Coventional -U Fermi-Hamiltonian R. Micnas et al., Rev. Mod. Phys. 62, 113 (1990).

SO coupled -U Fermi-Hamiltonian Q. Sun, G.-B. Zhu, W.-M. Liu, and A.-C. Ji, PRA 88, 063637 (2013) Two-body bound state solution

Results of SO coupled -U Fermi-Hamiltonian

Comments i). 在格点当中，SOC既能压制配对，也能增强配对。这是SOC与Lattice联合作用的结果，跟连续体系不同。 ii). 随着掺杂的增加，体系从Dirac cone主导的物理演变为SOC诱导的BCS-BEC crossover物理。

2) SO coupled +U Hubbard model For large U limit, , implement second-order perturbation: Dzyaloshinskii-Moriya (DM) interactions W. S. Cole et al., PRL 109, 085302 (2012) J. Radic et al., PRL 109, 085303 (2012)

W. S. Cole et al., PRL 109, 085302 (2012) J. Radic et al., PRL 109, 085303 (2012)

Superfluid-Mott insulator Transitions of Bosons in SO Coupled Optical Lattices Liang He, An-Chun Ji, and Walter Hofstetter, to appear in PRA (2015) Spin configuration：

Phase diagram of the Superfluid-Mott insulator Transitions HSF: homogenous superfluid

The effective Hamiltonian of chargon Slave-boson Theory chargon Introduce spinon constraint The effective Hamiltonian of chargon

Metal-Mott insulator Transitions of Fermions in SO Coupled Optical Lattices X. Zhang, W. Wu, G. Li, L. Wen, Q. Sun, and A.-C. Ji, New journal of Physics 17, 073036 (2015) SO coupled Fermi-Hubbard model:

Cluster Dynamical mean-Field theory (CDMFT) Weiss function of the cluster (1) Dyson equation: (2) Calculation procedure: impurity solver (1) (2)

Phase diagram in the whole parameter regime X. Zhang, W. Wu, G. Li, L. Wen, Q. Sun, and A.-C. Ji, New journal of Physics 17, 073036 (2015)

\alpha=0 \alpha=1.0 \alpha=1.0 \alpha=1.5

Drawback：need further size scaling! Non-magnetic insulator: candidate of algebraic spin liqiud Drawback：need further size scaling!

Quantum spin liquid emerging in 2D fermion system? Z. Y. Meng et al., Nature 464, 847 (2010)

Summary i). 双层体系是实现2D Rashba SOC 一个很好的 candidate，可以研究2D 自旋轨道耦合的玻色子和费米子多体效应。 ii). 对于光晶格体系，自旋轨道耦合的Hubbard模型具有丰富的强关联物理。

Thank you !